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A projective plane in $${\mathbb{R}}^ 4$$ with three critical points is standard. Strongly invertible knots have property P. (English) Zbl 0678.57003
The main result of this paper is that if one gets the unknot by attaching a band to the unknot, then the band is a trivial band with a half twist. The proof uses variations on the combinatorics of the intersection of planar surfaces developed by the second author [Invent. Math. 79, 125-141 (1985; Zbl 0559.57019) and ibid. 82, 37-55 (1985; Zbl 0576.57004)] and which have turned out to be very important elsewhere in 3-manifold topology - e.g. in the solution of the knot complement problem and in the cyclic surgery theorem. The theorems of the title are derived as consequences.
Reviewer: J.Hempel

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57R40 Embeddings in differential topology 57R70 Critical points and critical submanifolds in differential topology
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